While obtaining the highest possible data fit was not the primary objective of building our model, it may be useful to take a look at two standard outputs of the estimation procedure obtained with the Kalman filter.
First, by applying a two-sided smoothing of the data we can obtain the time series for historical stochastic disturbances, identified given the model structure. These are plotted in Figure 17.
A first look at the graphs confirms what we already stressed before: the magnitude of shocks hitting Poland is substantially larger than the size of shocks identified for the euro area. Focusing our attention on Poland, the estimates suggest that productivity in the tradable sector was relatively high in the middle of our sample and remained below average in the last three years. Roughly speaking, the opposite holds true for productivity in the nontradable sector and government spending shocks. Consumption preference and investment efficiency are clearly higher in the first part of our sample.21 Labour supply and monetary shocks do not exhibit such clear patterns, although the latter suggests
20 It may be argued that the negative response of private consumption to an increase in government
spending is inconsistent with the empirical evidence, indicating a positive comovement of both categories. In a DSGE framework, this effect can be obtained by assuming that a sufficiently large share of consumers behaves in a non-Ricardian way (see e.g. Gali et al., 2007). A less popular alternative is to allow for so-called deep habits in government spending (see Ravn et al., 2006).
21 Consumption preference and investment efficiency shocks in our model have their relatively close
counterparts in SOE-PL, which is a small open economy model developed by Adolfson et al. (2005) and estimated on the Polish data by Grabek et al. (2007). It is quite reassuring that the evolution in time of these disturbances roughly coincides in the two models.
4
a somewhat tighter monetary approach in the middle of our sample compared to more recent years. Given the unobservable nature of the stochastic processes, assessing the plausibility of their identification always involves a great deal of subjectivity. Therefore, we stop at this stage and do not attempt to link the evolution of the shocks to selected events and processes documented for the two economies. The second popular and simple tool to assess the empirical performance of a DSGE model is to cast its one-side Kalman filter predictions of the observable variables against their realizations (see Figure 18).
Figure 17. Stochastic disturbances
Concentrating again on the results for Poland, our model does a good job at tracking interest rates and inflation. Clearly, the model fails to account for the sharp and persistent
0 5 10 15 20 25 30 35 40 45 0 5 10 15 20 25 30 35 40 45 0 5 10 15 20 25 30 35 40 45 0 5 10 15 20 25 30 35 40 45 0 5 10 15 20 25 30 35 40 45 0 5 10 15 20 25 30 35 40 45 0 5 10 15 20 25 30 35 40 45 0 5 10 15 20 25 30 35 40 45 0 5 10 15 20 25 30 35 40 45 0 5 10 15 20 25 30 35 40 45 0 5 10 15 20 25 30 35 40 45 0 5 10 15 20 25 30 35 40 45 0 5 10 15 20 25 30 35 40 45 0 5 10 15 20 25 30 35 40 45 -20 0 20
Productivity shock in tradables - Poland
-20 0 20
Productivity shock in tradables - euro area
-10 0
10 Productivity shock in nontradables - Poland
-10 0
10 Productivity shock in nontradables - euro area
-20 0 20
Consumption preference shock - Poland
-20 0 20
Consumption preference shock - euro area
-20 0 20
Labour supply shock - Poland
-20 0 20
Labour supply shock - euro area
-20 0 20
Government spending shock - Poland
-20 0 20
Government spending shock - euro area
-20 0 20
Investment efficiency shock - Poland
-20 0 20
Investment efficiency shock - euro area
-1 0
1 Monetary policy shock - Poland
-1 0
1 Monetary policy shock - euro area 0 5 10 15 20 25 30 35 40 45 0 5 10 15 20 25 30 35 40 45 0 5 10 15 20 25 30 35 40 45 0 5 10 15 20 25 30 35 40 45 0 5 10 15 20 25 30 35 40 45 0 5 10 15 20 25 30 35 40 45 0 5 10 15 20 25 30 35 40 45 0 5 10 15 20 25 30 35 40 45 0 5 10 15 20 25 30 35 40 45 0 5 10 15 20 25 30 35 40 45 0 5 10 15 20 25 30 35 40 45 0 5 10 15 20 25 30 35 40 45 0 5 10 15 20 25 30 35 40 45 0 5 10 15 20 25 30 35 40 45 -20 0 20
Productivity shock in tradables - Poland
-20 0 20
Productivity shock in tradables - euro area
-10 0
10 Productivity shock in nontradables - Poland
-10 0
10 Productivity shock in nontradables - euro area
-20 0 20
Consumption preference shock - Poland
-20 0 20
Consumption preference shock - euro area
-20 0 20
Labour supply shock - Poland
-20 0 20
Labour supply shock - euro area
-20 0 20
Government spending shock - Poland
-20 0 20
Government spending shock - euro area
-20 0 20
Investment efficiency shock - Poland
-20 0 20
Investment efficiency shock - euro area
-1 0
1 Monetary policy shock - Poland
-1 0
4
slowdown in the economy in the middle of our sample and to capture volatility in real wagegrowth. All in all, the overall in-sample fit of our model seems to be acceptable if one takes into account a highly restrictive nature of the DSGE framework. It is worth noting that the empirical performance of our model seems to be significantly better in the last years of the sample. This is particularly true for GDP, consumption and real wage growth.
Figure 18. Data and one-step-ahead forecasts
Note: green line – data, brown dashed line – one-step-ahead forecast.
0 5 10 15 20 25 30 35 40 45 0 5 10 15 20 25 30 35 40 45 0 5 10 15 20 25 30 35 40 45 0 5 10 15 20 25 30 35 40 45 0 5 10 15 20 25 30 35 40 45 0 5 10 15 20 25 30 35 40 45 0 5 10 15 20 25 30 35 40 45 0 5 10 15 20 25 30 35 40 45 0 5 10 15 20 25 30 35 40 45 0 5 10 15 20 25 30 35 40 45 0 5 10 15 20 25 30 35 40 45 0 5 10 15 20 25 30 35 40 45 0 5 10 15 20 25 30 35 40 45 0 5 10 15 20 25 30 35 40 45 -2 0 2 GDP growth - Poland -1 0 1
GDP growth - euro area
-2 0 2
Consumption growth - Poland
-1 0 1
Consumption growth - euro area
-10 0 10
Investment growth - Poland
-2 0 2
Investment growth - euro area
-2 0 2
Real wage growth - Poland
-1 0 1
Real wage growth - euro area
-2 0 2
Internal exchange rate change - Poland
-1 0
1 Internal exchange rate change - euro area
-2 0 2
Interest rate - Poland
-1 0
1 Interest rate - euro area
-2 0 2 Inflation - Poland -1 0
1 Inflation - euro area 0 5 10 15 20 25 30 35 40 45 0 5 10 15 20 25 30 35 40 45 0 5 10 15 20 25 30 35 40 45 0 5 10 15 20 25 30 35 40 45 0 5 10 15 20 25 30 35 40 45 0 5 10 15 20 25 30 35 40 45 0 5 10 15 20 25 30 35 40 45 0 5 10 15 20 25 30 35 40 45 0 5 10 15 20 25 30 35 40 45 0 5 10 15 20 25 30 35 40 45 0 5 10 15 20 25 30 35 40 45 0 5 10 15 20 25 30 35 40 45 0 5 10 15 20 25 30 35 40 45 0 5 10 15 20 25 30 35 40 45 -2 0 2 GDP growth - Poland -1 0 1
GDP growth - euro area
-2 0 2
Consumption growth - Poland
-1 0 1
Consumption growth - euro area
-10 0 10
Investment growth - Poland
-2 0 2
Investment growth - euro area
-2 0 2
Real wage growth - Poland
-1 0 1
Real wage growth - euro area
-2 0 2
Internal exchange rate change - Poland
-1 0
1 Internal exchange rate change - euro area
-2 0 2
Interest rate - Poland
-1 0
1 Interest rate - euro area
-2 0 2 Inflation - Poland -1 0
5
5
Testing sources of heterogeneity
In order to assess the degree of heterogeneity between Poland and the euro area in a more formal way, we estimate several restricted versions of our model and compare it to its fully- fledged, unrestricted specification. As indicated in the introduction, the Bayesian approach to estimation provides a natural platform for comparisons across potentially misspecified models.22 Formally, this can be done by assigning prior probabilities to competing models and then using the Bayes’theorem to see how probable each model is given the data. Taking the ratio of the posterior probability for a given model to that of the reference model gives the posterior odds on the latter. In practice, the prior probabilities of each competing model are often assumed to be the same, in which case the posterior odds reduce to the Bayes factor, defined as the ratio of marginal likelihoods (see formula (60)) of the competing models.23
Calculating the marginal likelihood of a model is far from straightforward. There are two popular approaches to this problem (see Schorfheide, 2002). First, one can assume that the posterior kernel shape is close to normal, which yields the so-called Laplace approximation. The other method, typically referred to as the harmonic mean estimator, relies on simulating the marginal density function using the algorithm developed by Geweke (1999). The clear advantage of the former technique is its computational efficiency: all what is needed is the posterior maximization, while the harmonic mean estimator needs running the time-consuming Metropolis-Hastings algorithm. Since the number of restrictions we want to test is rather large, we calculate the marginal likelihood of various versions of our model using the Laplace approximation.24
Our strategy to testing the sources of heterogeneity between Poland and the euro area can be described as follows. In the first step, we test for cross country differences parameter by parameter.25 Next we consider several tests of multiple hypotheses. The results are reported in Table 10.
In what follows, we base our inference on the scheme suggested by Kass and Raftery
ZKLFKLVDPRGLILFDWLRQRIFODVVLFDOUXOHVODLGGRZQE\-HIIUH\V,QSDUWLFXODU
if the Bayes factor with respect to the unrestricted model is lower than 1/3 (1/20, 1/150), we treat it as positive (strong, very strong) evidence for heterogeneity between Poland and the euro area. It has to be noted that the marginal likelihood penalizes the model fit by a measure of its complexity. It means that it is perfectly possible that a restricted model will score better compared to its unrestricted version. In this case, the evidence in favour of homogeneity is judged in a symmetric fashion to that described above, with the cut-off points at 3, 20 and 150.
22 See e.g. Landon-Lane (1998).
23 Since the marginal likelihood of a model is directly related to the predictive density of the model, it
provides a natural platform for model comparisons based on the data fit. See e.g. Lancaster (2004).
24 As a robustness check, we assess the marginal data density for our key restricted specifications using the
harmonic mean estimator. The conclusions one can draw from comparing them to the unrestricted version are qualitatively the same as those obtained using the Laplace approximation.
25 Testing for perfect synchronization of shocks cannot be done in a straightforward manner, since
imposing a unity restriction on any cross correlation effectively reduces the number of shocks, which leads to a well-known problem of stochastic singularity, as long as all observable variables are used in the estimation. To deal with this problem, we approximate the synchronized versions of our model by setting cross correlations of the relevant shocks to 0.95.
5
Table 10. Sources of heterogeneity
No. Hypothesis Log marginal data
density Bayes factor 0 unrestricted model -439.7 1.000 1 ]=] -440.6 0.426 2 = -438.9 2.315 3 = -439.7 1.039 4 H’’=H’’ -438.8 2.489 5 == ; -438.3 4.033 6 C=C -437.2 12.944 7 L= L -439.2 1.673 8 ==; -437.7 7.568 9 C= C -439.1 1.756 10 L= L -440.8 0.354 11 = -444.6 0.008 12 n= n -441.4 0.188 13 = -439.2 1.633 14 V==V; -440.3 0.532 15 VC=VC -439.9 0.857 16 Y= Y -439.5 1.278 17 a= a -438.8 2.385 18 \=\ -438.4 3.872 19 ^= ^ -439.4 1.323 20 V== V; -444.5 0.008 21 VC= VC -442.2 0.085 22 Y=Y -445.3 0.004 23 a= a -441.2 0.227 24 \=\ -486.4 0.000 25 ^= ^ -449.4 0.000 26 b= b -461.0 0.000 27 XdggV=,; 1 -493.1 0.000 28 XdggVC1 -510.1 0.000 29 XdggY1 -483.8 0.000 30 Xdgga 1 -476.2 0.000 31 Xdgg\1 -487.8 0.000 32 Xdgg^ 1 -473.1 0.000 33 Xdggb 1 -492.7 0.000 34 utility (1–3) -439.7 1.007 35 price and wage formation (5–10) -434.6 174.469 36 structural (1–10) -433.8 380.615 37 policy reaction (11–13) -438.9 2.320 38 shock inertia (14–19) -437.6 8.536 39 shock volatility (20–26) -527.3 0.000 40 shock inertia and volatility (37–38) -543.6 0.000 41 (nearly) perfect correlation of shocks (27–33) -9.2E+18 0.000 42 stochastic homogeneity (39–41) -38034.8 0.000 43 no correlation of shocks -443.0 0.036
5
Given these inference rules and starting from simple hypotheses, we do not find strong evidence neither for nor against homogeneity in structural parameters and shock inertia coefficients. The only exception is the degree of interest rate smoothing, which is significantly different between Poland and the euro area. This is mostly due to the fact that these parameters are estimated with relatively low precision. On the contrary, our results speak strongly or very strongly in favour of heterogeneity in volatilities of most shock processes. Only in the case of nontradable sector productivity and labour supply shocks is the evidence for heterogeneity weak, despite substantial differences in point estimates. Finally, all simple hypotheses of nearly perfect cross-country shock correlations are very strongly rejected by the data.
Turning to multiple hypotheses, our estimated Bayes factors are consistent with structural homogeneity between Poland and the euro area: the model restricting the structural parameters (excluding the parametrization of the monetary policy reaction function) to be equal in both economies fits the data significantly better than a model assuming that all of them are different. This is to large extent because the structural parameters are estimated with a relatively low precision. Given that the tightness of our posterior distributions does not deviate much from other approaches to estimating DSGE models, including those using data for developed economies, a more general and somewhat pessimistic conclusion is that there seems to be relatively little information on the so-called deep parameters in standard macro-variables.
Despite strong evidence in favour of heterogeneity between interest rate smoothing pointed out above, the model assuming identical monetary policy feedback rules in Poland and in the euro area turns out to be as good as our baseline specification. Our results speak strongly against full stochastic homogeneity: shock volatilities differ across countries in a significant way and they are very far from being perfectly synchronized. It has to be noted that lack of cross correlation of stochastic disturbances is rejected by the data as well, although not as strongly as the perfect correlation hypothesis. Hence, our results should be viewed as suggesting moderate interdependence of shocks between Poland and the euro area.
6
6
Conclusions
The main objective of this paper was to build a two-country model linking Poland and the euro area and to apply it for assessment of heterogeneity across these two regions. Overall, our results can be seen as rather inconclusive about the differences in parameters driving behaviour of agents in Poland and the euro area. On the contrary, we find strong evidence for heterogeneity in terms of volatility and synchronization of shocks hitting both economies.
Our results suggest that a policy optimal for the euro area might not be optimal for Poland. This means that Poland’s entry to the EMU will involve costs associated with losing the monetary autonomy and stabilizing movements of the exchange rate. It is somewhat reassuring, however, that the detected extent of heterogeneity in terms of imperfect cross country correlation of stochastic disturbances is not very different from that obtained in studies covering relatively closely integrated euro area member states.
It should also be noted that the welfare losses associated with imperfect synchronization of shocks are usually found to be of small magnitude (see e.g. Corsetti, 2008). Nevertheless, a careful examination of such costs definitely warrants attention. A flexible design of our model makes it a good workhorse for comparing alternative monetary regimes, including the fixed exchange rate regime. We leave these interesting questions for future research.
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